The problem of designing an optimal quantizer with a fixed number of levels for a wide class of error weighting functions and an arbitrary distribution function is discussed. The existence of an optimal quantizer is proved, and a two-stage algorithm for its design is suggested. In this algorithm, at the first stage, Lloyd's iterative Method I is applied for reducing the region where, at the second stage, the search for an optimal quantizer is performed using a hybrid of the dynamic programming algorithm and the Lloyd-Max algorithm, which achieves the absolute optimality of dynamic programming with much less computational effort. For a continuous distribution with log-concave density and an increasing convex weighting function of the absolute quantization error, a reliable method is presented to compute the parameters of the optimal quantizer with a known precision using a generalization either of Lloyd's Method I or of the Lloyd-Max algorithm.>

Kieffer recently showed that for a log-concave probability density there is a uniqueN-level quantizer which is a stationary point of the expected value of an increasing, convex, and continuously differentiable error weighting function, dependent on the absolute quantization error. It is shown that the requirement of continuous differentiability can be dropped, using the monotony of Lloyd's Method II, the proof of which is based on a generalization of previous work by the author.

Sufficient conditions are presented for uniqueness of a locally optimal quantizer being a stationary point of the quantization distortion measureE[f(x,\eta)], the expected value of an error weighting functionf(x,\eta), wherexis a random variable to be quantized, where the probability density functionp(x)describingxis continuous and positive on some finite or infinite interval and zero outside it, and where\etais the quantization error. The functionf(x,\eta)is assumed convex and symmetric in\etaand zero only for\eta = 0. It is shown that in the cases off(x,\eta)=\eta^{2}andf(x,\eta)=|\eta|, the simple condition of concavity of Inp(x)is sufficient for uniqueness of a locally optimal quantizer.